. AY 



'-'' 



T A v 



ft MI) 



bMH 



\ :,',!, 



It ii not 



had n quarrel. 

 w Inn- in that 



-:eral 



used tfii part ien!;ir u'M lav lor th- 



lible tlmt Ubnliti 



ution. or if In- VMTC so. Sti - in rjifiiu 



wa* published in KiOSl vras not ;* hut Stevinns did not 

 tut any vcept tho-c ni li-u-s parallel to 



the ground, nor t'lialili neither: while Taylor did i/*r 

 thrill, which i- the di-tim-tive feature of his -\ stem. Airain. 

 it is ; in favour of Taylor's orifrinalitv 



in this point, tlmt works published abroad shortly ailer hi> 

 time do not contain it. For example. Hie K 

 Kmleitun;; zur 1'erspectiv, von J. ('. Bischott', 1711,' a 

 quarter of a century after the time of Taylor's publication, 

 contains no use of vanishing points except at the height of 

 the i 



The Mfthntlit* Iiicrementorwn is the first treatise in 

 which what U at this day called the calculus of finite dif- 

 ferences is, proposed for consideration. Besides what are 

 now the ni.>st common theorems in this subject, there are 

 various purely fluxional or infinitesimal theories, such as 

 the change of the independent variable, integrations, .T. 

 Bernoulli's series, Sec., and various applications to inter- 

 polation. the vibrating chord, the catenary, dome, tec.. 

 centre of oscillation and percussion, law of density of the 

 atmosphere. refraction of li;;ht. The first enunciation of 

 the celebrated theorem is as follows: 



PROP. VII. THEOR. III. 



Sint x et x quantitaten duie variabilcs. (juanim z unifor- 

 miter aupetur per data incremonta *, ct sit n: = r, 

 r z=Y, V r "/, et sic porro. Turn Hieo (mod quo 

 tem|x>re : crescendo fit c+r, :r item crescendo fiet 



Corollary I. expresses the corresponding theorem for decre- 

 ments. 



CdHOLL. II. 



Si pro Incrcmeiitis evaiiescentihus scrihantur (luxiones 



ipsis proportionales, factis jam omnibus V V, V, r, ,r, ,,r. 



i ijnalibus quo tempore z uniformiter fluendo fit 



x +.r r + .r 



U 



r- &c. 



vel mutato signo ipsius r, quo tempore s decrescendo fiel 

 x r, T decrescendo fiet 



.. .*. *. 



-,&c. 



Taylor does not make much ue of his own theorem in the 



Mel /i 'iit'iit'inim. but lie shows bis command over 



it in the paper above cited on the root* of equations, ii 

 which he extends Newton's method to other than alge- 

 braical equations. 



One would have supposed that such a theorem ax thai 



of Taylor, the instant it was proposed, would have been 



hailcJ a- the best and most useful of generalization*. In- 



I of this, it sunk, or rather nev .11 I.nfrransrt 



pointed out its power. This is perhaps an assertion whiel 



some may doubt : we proceed to make it pood. The first 



ciltiei-m upon the whole work (without a word nbout 



in was that ol Leibnitz, in u letter to John 



>ulli June. 17H>, vol. ii., p. 38(1, of their corrcspon- 



i what sort of view t' 



.-nc''. The Iran 



iufi?"""'" : '' ' ed what Taylor calls his Method 



the fac'i"'"' 1111 '"'^ ^' ' s an "J'l'h'cation of the differential and 

 ^1 calculus to iiiniiOKrt, or rutfu'r In ^nn nil MH<'- 

 -ain ' " 1US ''"' ''"tflitli have placed the horses. B 



el the |i.ovcil>. behind the cart. 1 heiran the clirl'e- 

 alcultui from series of numbers .... and so came 

 uerid calculus to the special gcome- 

 intimtesiiiml calculus. They proceed the other 



inn in a 



e 

 ' 



r,,^' ;'iey ha\r not the true method ol 



>/!/'''</ , It i> written obscurely enough.' Hernmilli on- 



'>/! 



' '"/// ''" 

 - J' 1 



' Si adumhraixlip |arallcl icrUf pur vilrcura 



inlnr. niarum umlinr <v>..lt'MtUr rnnnirn-iit in 



.<!> n*l pumllr^l: ! tdOAlbrmBiU! pivlmrntn 



m n-nmu t*tim lutudln* mpn |mviaminni mint 



sers (August, 17K ; , I have at lentrth r 



Taylor's book. What, in the n:i 



neau by the darkness in uhich hr m\u]\ 

 hint's!' No dc 

 far IL-, I can mnl- 



-tolen from me, through hi- Tlie 



lotion of I.eibnil/ p: 



our own day, ' 

 



n which pure a' and 



phj-ii-. and even a irem : f existing then 



ed in the lanjjuaire of that Calculus, was a ; 

 lively errotieou- mod. line. 



In Hritain, two 

 i. STIRI.IM. 



. 102 repeated the theorem as gi 

 adds that Herman h:i'l 



; and a> ' I7'<i. 



1C, llelllli, 



considered an independent in\c 

 the apjiendix to the Ph'ir/m 



we' find only the theorem in 



r Principja, and John Bernoulli's series for integration. 

 Maclaurin ' I'lin-inm, 17-12, p. 010 proved Taylor's theo- 

 111 the way which has since become common. 

 But both Stirling and Maclaurin use only a partictilai 

 of Taylor's theorem, expanding not ^ ('./ + ; >-)-; . 



or expanding <>; in po N. it!:er thuiiu-lit h 



doing more than proving Taylor's th- :illn- 



bute the result to Tavlor. Never! b .-iilur 



ca*e has been since called Mac'aiiiiii . h, if 



not Taylor's, it is Stirling's. Macl.iurin'.s booK 

 doubt, more read than either of the other two: it w:- 

 answer to Berkeley's metaphysical objections, ami 

 tsiined i;reat. power and vast store ol : and this 



may have been the reason why a theorem which wa- 

 in, and best known by, Maclfturin's book, shoi: 

 called after his name. It is well that it should be so, or 

 rather, it would be well that the development of i 

 in powers of : should be called by the name of Stirling : 

 for in truth the development of f ," + ('''/ in powers of Ii is 

 one theorem or another in its n.-es, and in the eon-eijneiices 

 . according as a or b is looked at a.s the principal 



In the interval between Taylor's death and I 

 paper in the Berlin Memoirs for 177-, in which he first 

 proposed to make Taylor's theorem the foundation of the 

 Differentia] Cah-uhis, the the: hardly known, 



and even when known, not known : 

 not find it in II Fluxions (1736), in 



nesi's Institution-. 171^ . in I-amlc- 

 1701 , in Simpson's Fluxions (1737. in Kmerson'k Iii- 

 7'!:t . in Kmcrxm's Fluxio 



'ictionaiy 174.'t , nor in the first edition 

 of Montucla's History 'l7"' . ^'. 



other places in which it should be, without 

 anywhere, except in the iri<;i1 French Encvclopsedu 

 tide ' certainly did tiiul it, mentioned 



only incidentally, and attrihut. * than 



net to D'Alenibert. Tli ;v ho wrote 



the jircliiniuaiy i 



'ime; though J . when he published Ins 



. of mathematics, he was better informed. \\Y 



.\vards) that Cundorcr n ..p.:'.ic 



wa.s in the habit g this the :cm to ]>. Member! : 



not with any unfair intention, hut in pure i The 



bert (Krc/ifn-/n:\ .\i/r 

 i. aci-ordin;: to 1 



'n-d h\ a method of tindini; tht 



it of T;i\ ' lain number of terms 



have 1 had never 



In fact. n'Alcmherl 1 if it uerc 



;uul without me- ' 'lie. which 



: : an tipinion in which we 



cannot agree. 1'nl. n^'lish, we can- 



not imagine how lie should have known Tav lor's theorem, 

 nor even then, unlex, Taylor. Stii bnir. Maelaurin. or an old 

 volumi' of the rhilo-o]ibii-al Ti ansiiet ion-. -ed to 



illen in his way. \Ve hav < no doidil that D'Alenibert. 

 wa a nW discoverer Of the tbeoiem. jmd that ''..udoieet 

 . pi in his writint's. Our wonder rather is 

 where Lapranee could have found the name of Taylor in 



